spherical_advection_diffusion_elements.h

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// LIC// ====================================================================
// LIC// This file forms part of oomph-lib, the object-oriented,
// LIC// multi-physics finite-element library, available
// LIC// at http://www.oomph-lib.org.
// LIC//
// LIC// Copyright (C) 2006-2023 Matthias Heil and Andrew Hazel
// LIC//
// LIC// This library is free software; you can redistribute it and/or
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// LIC// License as published by the Free Software Foundation; either
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// LIC// Lesser General Public License for more details.
// LIC//
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// LIC//====================================================================
// Header file for advection diffusion elements in a spherical polar coordinate
// system
#ifndef OOMPH_SPHERICAL_ADV_DIFF_ELEMENTS_HEADER
#define OOMPH_SPHERICAL_ADV_DIFF_ELEMENTS_HEADER
 
 
// Config header generated by autoconfig
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif
 
// OOMPH-LIB headers
#include "../generic/nodes.h"
#include "../generic/Qelements.h"
#include "../generic/refineable_elements.h"
#include "../generic/oomph_utilities.h"
 
namespace oomph
{
  //=============================================================
  /// A class for all elements that solve the
  /// Advection Diffusion equations in a spherical polar coordinate system
  /// using isoparametric elements.
  /// \f[ Pe \mathbf{w}\cdot(\mathbf{x}) \nabla u = \nabla \cdot \left( \nabla u \right) + f(\mathbf{x}) \f]
  /// This contains the generic maths. Shape functions, geometric
  /// mapping etc. must get implemented in derived class.
  //=============================================================
  class SphericalAdvectionDiffusionEquations : public virtual FiniteElement
  {
  public:
    /// Function pointer to source function fct(x,f(x)) --
    /// x is a Vector!
    typedef void (*SphericalAdvectionDiffusionSourceFctPt)(
      const Vector<double>& x, double& f);
 
 
    /// Function pointer to wind function fct(x,w(x)) --
    /// x is a Vector!
    typedef void (*SphericalAdvectionDiffusionWindFctPt)(
      const Vector<double>& x, Vector<double>& wind);
 
 
    /// Constructor: Initialise the Source_fct_pt and Wind_fct_pt
    /// to null and set (pointer to) Peclet number to default
    SphericalAdvectionDiffusionEquations() : Source_fct_pt(0), Wind_fct_pt(0)
    {
      // Set pointer to Peclet number to the default value zero
      Pe_pt = &Default_peclet_number;
      PeSt_pt = &Default_peclet_number;
    }
 
    /// Broken copy constructor
    SphericalAdvectionDiffusionEquations(
      const SphericalAdvectionDiffusionEquations& dummy) = delete;
 
    /// Broken assignment operator
    void operator=(const SphericalAdvectionDiffusionEquations&) = delete;
 
    /// Return the index at which the unknown value
    /// is stored. The default value, 0, is appropriate for single-physics
    /// problems, when there is only one variable, the value that satisfies
    /// the spherical advection-diffusion equation.
    /// In derived multi-physics elements, this function should be overloaded
    /// to reflect the chosen storage scheme. Note that these equations require
    /// that the unknown is always stored at the same index at each node.
    virtual inline unsigned u_index_spherical_adv_diff() const
    {
      return 0;
    }
 
 
    /// du/dt at local node n.
    /// Uses suitably interpolated value for hanging nodes.
    double du_dt_spherical_adv_diff(const unsigned& n) const
    {
      // Get the data's timestepper
      TimeStepper* time_stepper_pt = this->node_pt(n)->time_stepper_pt();
 
      // Initialise dudt
      double dudt = 0.0;
      // Loop over the timesteps, if there is a non Steady timestepper
      if (!time_stepper_pt->is_steady())
      {
        // Find the index at which the variable is stored
        const unsigned u_nodal_index = u_index_spherical_adv_diff();
 
        // Number of timsteps (past & present)
        const unsigned n_time = time_stepper_pt->ntstorage();
 
        for (unsigned t = 0; t < n_time; t++)
        {
          dudt +=
            time_stepper_pt->weight(1, t) * nodal_value(t, n, u_nodal_index);
        }
      }
      return dudt;
    }
 
 
    /// Disable ALE -- empty overload to suppress warning.
    /// ALE isn't implemented anyway
    void disable_ALE() {}
 
    /// Output with default number of plot points
    void output(std::ostream& outfile)
    {
      unsigned nplot = 5;
      output(outfile, nplot);
    }
 
    /// Output FE representation of soln: r,z,u  at
    /// nplot^2 plot points
    void output(std::ostream& outfile, const unsigned& nplot);
 
 
    /// C_style output with default number of plot points
    void output(FILE* file_pt)
    {
      unsigned n_plot = 5;
      output(file_pt, n_plot);
    }
 
    /// C-style output FE representation of soln: r,z,u  at
    /// n_plot^2 plot points
    void output(FILE* file_pt, const unsigned& n_plot);
 
 
    /// Output exact soln: r,z,u_exact at nplot^2 plot points
    void output_fct(std::ostream& outfile,
                    const unsigned& nplot,
                    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt);
 
    /// Get error against and norm of exact solution
    void compute_error(std::ostream& outfile,
                       FiniteElement::SteadyExactSolutionFctPt exact_soln_pt,
                       double& error,
                       double& norm);
 
    /// Access function: Pointer to source function
    inline SphericalAdvectionDiffusionSourceFctPt& source_fct_pt()
    {
      return Source_fct_pt;
    }
 
 
    /// Access function: Pointer to source function. Const version
    inline SphericalAdvectionDiffusionSourceFctPt source_fct_pt() const
    {
      return Source_fct_pt;
    }
 
 
    /// Access function: Pointer to wind function
    inline SphericalAdvectionDiffusionWindFctPt& wind_fct_pt()
    {
      return Wind_fct_pt;
    }
 
 
    /// Access function: Pointer to wind function. Const version
    inline SphericalAdvectionDiffusionWindFctPt wind_fct_pt() const
    {
      return Wind_fct_pt;
    }
 
    // Access functions for the physical constants
 
    /// Peclet number
    inline const double& pe() const
    {
      return *Pe_pt;
    }
 
    /// Pointer to Peclet number
    inline double*& pe_pt()
    {
      return Pe_pt;
    }
 
    /// Peclet number multiplied by Strouhal number
    inline const double& pe_st() const
    {
      return *PeSt_pt;
    }
 
    /// Pointer to Peclet number multipled by Strouha number
    inline double*& pe_st_pt()
    {
      return PeSt_pt;
    }
 
    /// Get source term at (Eulerian) position x. This function is
    /// virtual to allow overloading in multi-physics problems where
    /// the strength of the source function might be determined by
    /// another system of equations
    inline virtual void get_source_spherical_adv_diff(const unsigned& ipt,
                                                      const Vector<double>& x,
                                                      double& source) const
    {
      // If no source function has been set, return zero
      if (Source_fct_pt == 0)
      {
        source = 0.0;
      }
      else
      {
        // Get source strength
        (*Source_fct_pt)(x, source);
      }
    }
 
    /// Get wind at (Eulerian) position x and/or local coordinate s.
    /// This function is
    /// virtual to allow overloading in multi-physics problems where
    /// the wind function might be determined by
    /// another system of equations
    inline virtual void get_wind_spherical_adv_diff(const unsigned& ipt,
                                                    const Vector<double>& s,
                                                    const Vector<double>& x,
                                                    Vector<double>& wind) const
    {
      // If no wind function has been set, return zero
      if (Wind_fct_pt == 0)
      {
        for (unsigned i = 0; i < 3; i++)
        {
          wind[i] = 0.0;
        }
      }
      else
      {
        // Get wind
        (*Wind_fct_pt)(x, wind);
      }
    }
 
    /// Get flux:
    /// \f[ \mbox{flux}[i] = \nabla u = \mbox{d}u / \mbox{d} r + 1/r \mbox{d}u / \mbox{d} \theta \f]
    void get_flux(const Vector<double>& s, Vector<double>& flux) const
    {
      // Find out how many nodes there are in the element
      const unsigned n_node = nnode();
 
      // Get the nodal index at which the unknown is stored
      const unsigned u_nodal_index = u_index_spherical_adv_diff();
 
      // Set up memory for the shape and test functions
      Shape psi(n_node);
      DShape dpsidx(n_node, 2);
 
      // Call the derivatives of the shape and test functions
      dshape_eulerian(s, psi, dpsidx);
 
      // Initialise to zero
      for (unsigned j = 0; j < 2; j++)
      {
        flux[j] = 0.0;
      }
 
      // Loop over nodes
      for (unsigned l = 0; l < n_node; l++)
      {
        const double u_value = this->nodal_value(l, u_nodal_index);
        const double r = this->nodal_position(l, 0);
        // Add in the derivative directions
        flux[0] += u_value * dpsidx(l, 0);
        flux[1] += u_value * dpsidx(l, 1) / r;
      }
    }
 
 
    /// Add the element's contribution to its residual vector (wrapper)
    void fill_in_contribution_to_residuals(Vector<double>& residuals)
    {
      // Call the generic residuals function with flag set to 0 and using
      // a dummy matrix
      fill_in_generic_residual_contribution_spherical_adv_diff(
        residuals,
        GeneralisedElement::Dummy_matrix,
        GeneralisedElement::Dummy_matrix,
        0);
    }
 
 
    /// Add the element's contribution to its residual vector and
    /// the element Jacobian matrix (wrapper)
    void fill_in_contribution_to_jacobian(Vector<double>& residuals,
                                          DenseMatrix<double>& jacobian)
    {
      // Call the generic routine with the flag set to 1
      fill_in_generic_residual_contribution_spherical_adv_diff(
        residuals, jacobian, GeneralisedElement::Dummy_matrix, 1);
    }
 
    /// Add the element's contribution to its residual vector and
    /// the element Jacobian matrix (wrapper) and mass matrix
    void fill_in_contribution_to_jacobian_and_mass_matrix(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      DenseMatrix<double>& mass_matrix)
    {
      // Call the generic routine with the flag set to 2
      fill_in_generic_residual_contribution_spherical_adv_diff(
        residuals, jacobian, mass_matrix, 2);
    }
 
 
    /// Return FE representation of function value u(s) at local coordinate s
    inline double interpolated_u_spherical_adv_diff(
      const Vector<double>& s) const
    {
      // Find number of nodes
      const unsigned n_node = nnode();
 
      // Get the nodal index at which the unknown is stored
      const unsigned u_nodal_index = u_index_spherical_adv_diff();
 
      // Local shape function
      Shape psi(n_node);
 
      // Find values of shape function
      shape(s, psi);
 
      // Initialise value of u
      double interpolated_u = 0.0;
 
      // Loop over the local nodes and sum
      for (unsigned l = 0; l < n_node; l++)
      {
        interpolated_u += nodal_value(l, u_nodal_index) * psi[l];
      }
 
      return (interpolated_u);
    }
 
 
    /// Return derivative of u at point s with respect to all data
    /// that can affect its value.
    /// In addition, return the global equation numbers corresponding to the
    /// data. This is virtual so that it can be overloaded in the
    /// refineable version
    virtual void dinterpolated_u_adv_diff_ddata(
      const Vector<double>& s,
      Vector<double>& du_ddata,
      Vector<unsigned>& global_eqn_number)
    {
      // Find number of nodes
      const unsigned n_node = nnode();
 
      // Get the nodal index at which the unknown is stored
      const unsigned u_nodal_index = u_index_spherical_adv_diff();
 
      // Local shape function
      Shape psi(n_node);
 
      // Find values of shape function
      shape(s, psi);
 
      // Find the number of dofs associated with interpolated u
      unsigned n_u_dof = 0;
      for (unsigned l = 0; l < n_node; l++)
      {
        int global_eqn = this->node_pt(l)->eqn_number(u_nodal_index);
        // If it's positive add to the count
        if (global_eqn >= 0)
        {
          ++n_u_dof;
        }
      }
 
      // Now resize the storage schemes
      du_ddata.resize(n_u_dof, 0.0);
      global_eqn_number.resize(n_u_dof, 0);
 
      // Loop over the nodes again and set the derivatives
      unsigned count = 0;
      for (unsigned l = 0; l < n_node; l++)
      {
        // Get the global equation number
        int global_eqn = this->node_pt(l)->eqn_number(u_nodal_index);
        // If it's positive
        if (global_eqn >= 0)
        {
          // Set the global equation number
          global_eqn_number[count] = global_eqn;
          // Set the derivative with respect to the unknown
          du_ddata[count] = psi[l];
          // Increase the counter
          ++count;
        }
      }
    }
 
 
    /// Self-test: Return 0 for OK
    unsigned self_test();
 
  protected:
    /// Shape/test functions and derivs w.r.t. to global coords at
    /// local coord. s; return  Jacobian of mapping
    virtual double dshape_and_dtest_eulerian_spherical_adv_diff(
      const Vector<double>& s,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const = 0;
 
    /// Shape/test functions and derivs w.r.t. to global coords at
    /// integration point ipt; return  Jacobian of mapping
    virtual double dshape_and_dtest_eulerian_at_knot_spherical_adv_diff(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const = 0;
 
    /// Add the element's contribution to its residual vector only
    /// (if flag=and/or element  Jacobian matrix
    virtual void fill_in_generic_residual_contribution_spherical_adv_diff(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      DenseMatrix<double>& mass_matrix,
      unsigned flag);
 
    // Physical constants
 
    /// Pointer to global Peclet number
    double* Pe_pt;
 
    /// Pointer to global Peclet number multiplied by Strouhal number
    double* PeSt_pt;
 
    /// Pointer to source function:
    SphericalAdvectionDiffusionSourceFctPt Source_fct_pt;
 
    /// Pointer to wind function:
    SphericalAdvectionDiffusionWindFctPt Wind_fct_pt;
 
  private:
    /// Static default value for the Peclet number
    static double Default_peclet_number;
 
 
  }; // End class SphericalAdvectionDiffusionEquations
 
 
  /// ////////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////////
 
 
  //======================================================================
  /// QSphericalAdvectionDiffusionElement elements are
  /// linear/quadrilateral/brick-shaped Axisymmetric Advection Diffusion
  /// elements with isoparametric interpolation for the function.
  //======================================================================
  template<unsigned NNODE_1D>
  class QSphericalAdvectionDiffusionElement
    : public virtual QElement<2, NNODE_1D>,
      public virtual SphericalAdvectionDiffusionEquations
  {
  private:
    /// Static array of ints to hold number of variables at
    /// nodes: Initial_Nvalue[n]
    static const unsigned Initial_Nvalue;
 
  public:
    /// Constructor: Call constructors for QElement and
    /// Advection Diffusion equations
    QSphericalAdvectionDiffusionElement()
      : QElement<2, NNODE_1D>(), SphericalAdvectionDiffusionEquations()
    {
    }
 
    /// Broken copy constructor
    QSphericalAdvectionDiffusionElement(
      const QSphericalAdvectionDiffusionElement<NNODE_1D>& dummy) = delete;
 
    /// Broken assignment operator
    void operator=(const QSphericalAdvectionDiffusionElement<NNODE_1D>&) =
      delete;
 
    ///  Required  # of `values' (pinned or dofs)
    /// at node n
    inline unsigned required_nvalue(const unsigned& n) const
    {
      return Initial_Nvalue;
    }
 
    /// Output function:
    ///  r,z,u
    void output(std::ostream& outfile)
    {
      SphericalAdvectionDiffusionEquations::output(outfile);
    }
 
    /// Output function:
    ///  r,z,u  at n_plot^2 plot points
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      SphericalAdvectionDiffusionEquations::output(outfile, n_plot);
    }
 
 
    /// C-style output function:
    ///  r,z,u
    void output(FILE* file_pt)
    {
      SphericalAdvectionDiffusionEquations::output(file_pt);
    }
 
    ///  C-style output function:
    ///   r,z,u at n_plot^2 plot points
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      SphericalAdvectionDiffusionEquations::output(file_pt, n_plot);
    }
 
    /// Output function for an exact solution:
    ///  r,z,u_exact at n_plot^2 plot points
    void output_fct(std::ostream& outfile,
                    const unsigned& n_plot,
                    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
    {
      SphericalAdvectionDiffusionEquations::output_fct(
        outfile, n_plot, exact_soln_pt);
    }
 
 
  protected:
    /// Shape, test functions & derivs. w.r.t. to global coords. Return
    /// Jacobian.
    inline double dshape_and_dtest_eulerian_spherical_adv_diff(
      const Vector<double>& s,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const;
 
    /// Shape, test functions & derivs. w.r.t. to global coords. at
    /// integration point ipt. Return Jacobian.
    inline double dshape_and_dtest_eulerian_at_knot_spherical_adv_diff(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const;
 
  }; // End class QSphericalAdvectionDiffusionElement
 
  // Inline functions:
 
  //======================================================================
  /// Define the shape functions and test functions and derivatives
  /// w.r.t. global coordinates and return Jacobian of mapping.
  ///
  /// Galerkin: Test functions = shape functions
  //======================================================================
 
  template<unsigned NNODE_1D>
  double QSphericalAdvectionDiffusionElement<NNODE_1D>::
    dshape_and_dtest_eulerian_spherical_adv_diff(const Vector<double>& s,
                                                 Shape& psi,
                                                 DShape& dpsidx,
                                                 Shape& test,
                                                 DShape& dtestdx) const
  {
    // Call the geometrical shape functions and derivatives
    double J = this->dshape_eulerian(s, psi, dpsidx);
 
    // Loop over the test functions and derivatives and set them equal to the
    // shape functions
    for (unsigned i = 0; i < NNODE_1D; i++)
    {
      test[i] = psi[i];
      for (unsigned j = 0; j < 2; j++)
      {
        dtestdx(i, j) = dpsidx(i, j);
      }
    }
 
    // Return the jacobian
    return J;
  }
 
 
  //======================================================================
  /// Define the shape functions and test functions and derivatives
  /// w.r.t. global coordinates and return Jacobian of mapping.
  ///
  /// Galerkin: Test functions = shape functions
  //======================================================================
 
  template<unsigned NNODE_1D>
  double QSphericalAdvectionDiffusionElement<NNODE_1D>::
    dshape_and_dtest_eulerian_at_knot_spherical_adv_diff(const unsigned& ipt,
                                                         Shape& psi,
                                                         DShape& dpsidx,
                                                         Shape& test,
                                                         DShape& dtestdx) const
  {
    // Call the geometrical shape functions and derivatives
    double J = this->dshape_eulerian_at_knot(ipt, psi, dpsidx);
 
    // Set the test functions equal to the shape functions (pointer copy)
    test = psi;
    dtestdx = dpsidx;
 
    // Return the jacobian
    return J;
  }
 
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
 
  template<unsigned NNODE_1D>
  class FaceGeometry<QSphericalAdvectionDiffusionElement<NNODE_1D>>
    : public virtual QElement<1, NNODE_1D>
  {
  public:
    /// Constructor: Call the constructor for the
    /// appropriate lower-dimensional QElement
    FaceGeometry() : QElement<1, NNODE_1D>() {}
  };
 
 
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
 
  //======================================================================
  /// A class for elements that allow the imposition of an
  /// applied Robin boundary condition on the boundaries of Steady
  /// Axisymmnetric Advection Diffusion Flux elements.
  /// \f[ -\Delta u \cdot \mathbf{n} + \alpha(r,z) u = \beta(r,z) \f]
  /// The element geometry is obtained from the FaceGeometry<ELEMENT>
  /// policy class.
  //======================================================================
  template<class ELEMENT>
  class SphericalAdvectionDiffusionFluxElement
    : public virtual FaceGeometry<ELEMENT>,
      public virtual FaceElement
  {
  public:
    /// Function pointer to the prescribed-beta function fct(x,beta(x))
    /// -- x is a Vector!
    typedef void (*SphericalAdvectionDiffusionPrescribedBetaFctPt)(
      const Vector<double>& x, double& beta);
 
    /// Function pointer to the prescribed-alpha function fct(x,alpha(x))
    /// -- x is a Vector!
    typedef void (*SphericalAdvectionDiffusionPrescribedAlphaFctPt)(
      const Vector<double>& x, double& alpha);
 
 
    /// Constructor, takes the pointer to the "bulk" element
    /// and the index of the face to be created
    SphericalAdvectionDiffusionFluxElement(FiniteElement* const& bulk_el_pt,
                                           const int& face_index);
 
 
    /// Broken empty constructor
    SphericalAdvectionDiffusionFluxElement()
    {
      throw OomphLibError("Don't call empty constructor for "
                          "SphericalAdvectionDiffusionFluxElement",
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Broken copy constructor
    SphericalAdvectionDiffusionFluxElement(
      const SphericalAdvectionDiffusionFluxElement& dummy) = delete;
 
    /// Broken assignment operator
    void operator=(const SphericalAdvectionDiffusionFluxElement&) = delete;
 
    /// Access function for the prescribed-beta function pointer
    SphericalAdvectionDiffusionPrescribedBetaFctPt& beta_fct_pt()
    {
      return Beta_fct_pt;
    }
 
    /// Access function for the prescribed-alpha function pointer
    SphericalAdvectionDiffusionPrescribedAlphaFctPt& alpha_fct_pt()
    {
      return Alpha_fct_pt;
    }
 
 
    /// Add the element's contribution to its residual vector
    inline void fill_in_contribution_to_residuals(Vector<double>& residuals)
    {
      // Call the generic residuals function with flag set to 0
      // using a dummy matrix
      fill_in_generic_residual_contribution_spherical_adv_diff_flux(
        residuals, GeneralisedElement::Dummy_matrix, 0);
    }
 
 
    /// Add the element's contribution to its residual vector and
    /// its Jacobian matrix
    inline void fill_in_contribution_to_jacobian(Vector<double>& residuals,
                                                 DenseMatrix<double>& jacobian)
    {
      // Call the generic routine with the flag set to 1
      fill_in_generic_residual_contribution_spherical_adv_diff_flux(
        residuals, jacobian, 1);
    }
 
    /// Specify the value of nodal zeta from the face geometry
    /// The "global" intrinsic coordinate of the element when
    /// viewed as part of a geometric object should be given by
    /// the FaceElement representation, by default (needed to break
    /// indeterminacy if bulk element is SolidElement)
    double zeta_nodal(const unsigned& n,
                      const unsigned& k,
                      const unsigned& i) const
    {
      return FaceElement::zeta_nodal(n, k, i);
    }
 
    /// Output function -- forward to broken version in FiniteElement
    /// until somebody decides what exactly they want to plot here...
    void output(std::ostream& outfile)
    {
      FiniteElement::output(outfile);
    }
 
    /// Output function -- forward to broken version in FiniteElement
    /// until somebody decides what exactly they want to plot here...
    void output(std::ostream& outfile, const unsigned& nplot)
    {
      FiniteElement::output(outfile, nplot);
    }
 
 
  protected:
    /// Function to compute the shape and test functions and to return
    /// the Jacobian of mapping between local and global (Eulerian)
    /// coordinates
    inline double shape_and_test(const Vector<double>& s,
                                 Shape& psi,
                                 Shape& test) const
    {
      // Find number of nodes
      unsigned n_node = nnode();
 
      // Get the shape functions
      shape(s, psi);
 
      // Set the test functions to be the same as the shape functions
      for (unsigned i = 0; i < n_node; i++)
      {
        test[i] = psi[i];
      }
 
      // Return the value of the jacobian
      return J_eulerian(s);
    }
 
 
    /// Function to compute the shape and test functions and to return
    /// the Jacobian of mapping between local and global (Eulerian)
    /// coordinates
    inline double shape_and_test_at_knot(const unsigned& ipt,
                                         Shape& psi,
                                         Shape& test) const
    {
      // Find number of nodes
      unsigned n_node = nnode();
 
      // Get the shape functions
      shape_at_knot(ipt, psi);
 
      // Set the test functions to be the same as the shape functions
      for (unsigned i = 0; i < n_node; i++)
      {
        test[i] = psi[i];
      }
 
      // Return the value of the jacobian
      return J_eulerian_at_knot(ipt);
    }
 
    /// Function to calculate the prescribed beta at a given spatial
    /// position
    void get_beta(const Vector<double>& x, double& beta)
    {
      // If the function pointer is zero return zero
      if (Beta_fct_pt == 0)
      {
        beta = 0.0;
      }
      // Otherwise call the function
      else
      {
        (*Beta_fct_pt)(x, beta);
      }
    }
 
    /// Function to calculate the prescribed alpha at a given spatial
    /// position
    void get_alpha(const Vector<double>& x, double& alpha)
    {
      // If the function pointer is zero return zero
      if (Alpha_fct_pt == 0)
      {
        alpha = 0.0;
      }
      // Otherwise call the function
      else
      {
        (*Alpha_fct_pt)(x, alpha);
      }
    }
 
  private:
    /// Add the element's contribution to its residual vector.
    /// flag=1(or 0): do (or don't) compute the Jacobian as well.
    void fill_in_generic_residual_contribution_spherical_adv_diff_flux(
      Vector<double>& residuals, DenseMatrix<double>& jacobian, unsigned flag);
 
 
    /// Function pointer to the (global) prescribed-beta function
    SphericalAdvectionDiffusionPrescribedBetaFctPt Beta_fct_pt;
 
    /// Function pointer to the (global) prescribed-alpha function
    SphericalAdvectionDiffusionPrescribedAlphaFctPt Alpha_fct_pt;
 
    /// The index at which the unknown is stored at the nodes
    unsigned U_index_adv_diff;
 
 
  }; // End class SphericalAdvectionDiffusionFluxElement
 
 
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
 
 
  //===========================================================================
  /// Constructor, takes the pointer to the "bulk" element and the index
  /// of the face to be created
  //===========================================================================
  template<class ELEMENT>
  SphericalAdvectionDiffusionFluxElement<ELEMENT>::
    SphericalAdvectionDiffusionFluxElement(FiniteElement* const& bulk_el_pt,
                                           const int& face_index)
    : FaceGeometry<ELEMENT>(), FaceElement()
 
  {
    // Let the bulk element build the FaceElement, i.e. setup the pointers
    // to its nodes (by referring to the appropriate nodes in the bulk
    // element), etc.
    bulk_el_pt->build_face_element(face_index, this);
 
#ifdef PARANOID
    {
      // Check that the element is not a refineable 3d element
      ELEMENT* elem_pt = dynamic_cast<ELEMENT*>(bulk_el_pt);
 
      // If it's three-d
      if (elem_pt->dim() == 3)
      {
        // Is it refineable
        RefineableElement* ref_el_pt =
          dynamic_cast<RefineableElement*>(elem_pt);
        if (ref_el_pt != 0)
        {
          if (this->has_hanging_nodes())
          {
            throw OomphLibError("This flux element will not work correctly if "
                                "nodes are hanging\n",
                                OOMPH_CURRENT_FUNCTION,
                                OOMPH_EXCEPTION_LOCATION);
          }
        }
      }
    }
#endif
 
    // Initialise the prescribed-beta function pointer to zero
    Beta_fct_pt = 0;
 
    // Set up U_index_adv_diff. Initialise to zero, which probably won't change
    // in most cases, oh well, the price we pay for generality
    U_index_adv_diff = 0;
 
    // Cast to the appropriate AdvectionDiffusionEquation so that we can
    // find the index at which the variable is stored
    // We assume that the dimension of the full problem is the same
    // as the dimension of the node, if this is not the case you will have
    // to write custom elements, sorry
    SphericalAdvectionDiffusionEquations* eqn_pt =
      dynamic_cast<SphericalAdvectionDiffusionEquations*>(bulk_el_pt);
 
    // If the cast has failed die
    if (eqn_pt == 0)
    {
      std::string error_string =
        "Bulk element must inherit from SphericalAdvectionDiffusionEquations.";
      error_string +=
        "Nodes are two dimensional, but cannot cast the bulk element to\n";
      error_string += "SphericalAdvectionDiffusionEquations<2>\n.";
      error_string +=
        "If you desire this functionality, you must implement it yourself\n";
 
      throw OomphLibError(
        error_string, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
    else
    {
      // Read the index from the (cast) bulk element.
      U_index_adv_diff = eqn_pt->u_index_spherical_adv_diff();
    }
  }
 
 
  //===========================================================================
  /// Compute the element's residual vector and the (zero) Jacobian
  /// matrix for the Robin boundary condition:
  /// \f[ \Delta u \cdot \mathbf{n} + \alpha (\mathbf{x}) = \beta (\mathbf{x}) \f]
  //===========================================================================
  template<class ELEMENT>
  void SphericalAdvectionDiffusionFluxElement<ELEMENT>::
    fill_in_generic_residual_contribution_spherical_adv_diff_flux(
      Vector<double>& residuals, DenseMatrix<double>& jacobian, unsigned flag)
  {
    // Find out how many nodes there are
    const unsigned n_node = nnode();
 
    // Locally cache the index at which the variable is stored
    const unsigned u_index_spherical_adv_diff = U_index_adv_diff;
 
    // Set up memory for the shape and test functions
    Shape psif(n_node), testf(n_node);
 
    // Set the value of n_intpt
    const unsigned n_intpt = integral_pt()->nweight();
 
    // Set the Vector to hold local coordinates
    Vector<double> s(1);
 
    // Integers used to store the local equation number and local unknown
    // indices for the residuals and jacobians
    int local_eqn = 0, local_unknown = 0;
 
    // Loop over the integration points
    //--------------------------------
    for (unsigned ipt = 0; ipt < n_intpt; ipt++)
    {
      // Assign values of s
      for (unsigned i = 0; i < 1; i++)
      {
        s[i] = integral_pt()->knot(ipt, i);
      }
 
      // Get the integral weight
      double w = integral_pt()->weight(ipt);
 
      // Find the shape and test functions and return the Jacobian
      // of the mapping
      double J = shape_and_test(s, psif, testf);
 
      // Premultiply the weights and the Jacobian
      double W = w * J;
 
      // Calculate local values of the solution and its derivatives
      // Allocate
      double interpolated_u = 0.0;
      Vector<double> interpolated_x(2, 0.0);
 
      // Calculate position
      for (unsigned l = 0; l < n_node; l++)
      {
        // Get the value at the node
        double u_value = raw_nodal_value(l, u_index_spherical_adv_diff);
        interpolated_u += u_value * psif(l);
        // Loop over coordinate direction
        for (unsigned i = 0; i < 2; i++)
        {
          interpolated_x[i] += nodal_position(l, i) * psif(l);
        }
      }
 
      // Get the imposed beta (beta=flux when alpha=0.0)
      double beta;
      get_beta(interpolated_x, beta);
 
      // Get the imposed alpha
      double alpha;
      get_alpha(interpolated_x, alpha);
 
      // calculate the area weighting dS = r^{2} sin theta dr dtheta
      // r = x[0] and theta = x[1]
      double dS =
        interpolated_x[0] * interpolated_x[0] * sin(interpolated_x[1]);
 
      // Now add to the appropriate equations
 
      // Loop over the test functions
      for (unsigned l = 0; l < n_node; l++)
      {
        // Set the local equation number
        local_eqn = nodal_local_eqn(l, u_index_spherical_adv_diff);
        /*IF it's not a boundary condition*/
        if (local_eqn >= 0)
        {
          // Add the prescribed beta terms
          residuals[local_eqn] -=
            dS * (beta - alpha * interpolated_u) * testf(l) * W;
 
          // Calculate the Jacobian
          //----------------------
          if ((flag) && (alpha != 0.0))
          {
            // Loop over the velocity shape functions again
            for (unsigned l2 = 0; l2 < n_node; l2++)
            {
              // Set the number of the unknown
              local_unknown = nodal_local_eqn(l2, u_index_spherical_adv_diff);
 
              // If at a non-zero degree of freedom add in the entry
              if (local_unknown >= 0)
              {
                jacobian(local_eqn, local_unknown) +=
                  dS * alpha * psif[l2] * testf[l] * W;
              }
            }
          }
        }
      } // end loop over test functions
 
    } // end loop over integration points
 
  } // end fill_in_generic_residual_contribution_adv_diff_flux
 
 
} // namespace oomph
 
#endif